Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)	I. Linear Differential Equation of type \(\frac{d y}{d x}+P y=Q\)
B. \(\frac{d y}{d x}-2 y=\cos 3 x\)	II. Homogeneous Differential Equation
C. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	III. Linear Differential Equation of type \(\frac{d x}{d y}+P x=Q\)
D. \(\left(x+2 y^3\right) \frac{d y}{d x}=y\)	IV. Variable separable

Choose the correct answer from the options given below:

The point on the curve \(y^2=x\) where the tangent makes \(45^{\circ}\) angle with the \(x\)-axis is:

The corner points of the feasible region determined by the system of linear inequalities are \((0, 0), (0, 4), (4, 0), (2, 4)\) and \((0, 5)\). If the maximum value of \(Z = ax + by\) where \(a, b > 0\) occurs at both \((2, 4)\) and \((4, 0)\), then

Match \(List - I\) with \(List - II\)

\(List - I\)	\(List - II\)
(A) \(\frac{d}{d x}\left(\sin x^2\right)\)	(I) \(\frac{1}{5}\)
(B) \(\frac{d}{d x}\left(e^{\sin x}\right)\)	(II) 0
(C) \(f(x)=\tan ^{-1} x\) then \(f^{\prime}(2)\)	(III) \(2 x \cos x^2\)
(D) If \(y=3 \cos x-2 \sin x\), then \(\frac{d^2 y}{d x^2}+y\)	(IV) \(e^{\sin x} \cos x\)

If \(y=\sqrt{a x+b}\), then \(y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2=\)

If a function \(f: R _{+} \rightarrow[-5, \infty)\) given by \(f(x)=9 x^2+6 x-5\) is one-one and onto, then \(f^{-1}(y)\) is

The area (in sq. units) enclosed by the curve \(y=\frac{1}{2} x|x|\) and the line \(y=x\) is :

Answer the question on the basis of passage given below:
Proteins are the polymers of about twenty different a-amino acids which are linked by peptide bonds. Proteins perform various structural and dynamic functions in the organisms. Proteins which contain only a-amino acids are called simple proteins. The secondary or tertiary structure of proteins get disturbed on change of pH or temperature and they are not able to perform their functions. This is called denaturation of proteins. Enzymes are biocatalysts which speed up the reactions in biosystems. Vitamins are accessory food factors required in the diet. They are classified as fat soluble (A, D, E and K) and water soluble (B group and C). Deficiency of vitamins leads to many diseases. Nucleic acids are the polymers of nucleotides which in turn consist of a base, a pentose sugar and phosphate moiety. Nucleic acids are responsible for the transfer of characters from parents to offsprings. There are two types of nucleic acids DNA and RNA. DNA contains a five carbon sugar molecule called 2-deoxyribose whereas RNA contains ribose.
Which of the following is an essential amino acid?

Coagulating power of an ion for a colloidal solution depends on :

Alien DNA can be introduced in the animal host cell by :

The storage battery of a car has an emf of 10 V. If the internal resistance of the battery is \( 0.2 \Omega \), then the maximum current that can be drawn from the battery is

\(A\) and \(B\) take turns in throwing a pair of dice, the first to throw a sum of 9 wins the prize.
If \(A\) throws first, then the ratio of the probabilities of \(A\) and \(B\) winning is :

Select the hormone which is not secreted by human placenta: